# Systematic Workflow for Efficient Identification of Local Representative Elementary Volumes Demonstrated with Lithium-Ion Battery Cathode Microstructures

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Image Generation and Post-Processing

^{®}[27]. This software uses a neural network algorithm that has specifically been trained to distinguish pore space, active material, and the carbon-binder domain (CBD) in NMC-like battery electrode microstructures. A similar workflow for graphite-like battery electrode microstructures can be found in a recent related publication [40].

#### 2.2. Application of the lREV Workflow

- (1)
- General definitions

- (2)
- Extraction of Pore Network Information

- (3)
- Estimation of lower bound REV

- (4)
- Striding Windows

- (5)
- Penalty Function

#### 2.3. Simulation Methods

#### 2.3.1. Diffusion

#### 2.3.2. Hydrodynamics

#### 2.3.3. Electrochemistry

## 3. Results

- Step One: Extraction of Pore Network Information

- Step Two: Estimation of the lower bound of the REV

- Step Three: Striding Windows

- Step Four: Penalty Function

#### 3.1. Diffusion and Hydrodynamics Benchmark

#### 3.2. Electrochemical Benchmark

## 4. Discussion

## 5. Conclusions and Outlook

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AM | Active material |

CBD | Carbon-binder domain |

CFD | Computational fluid dynamics |

CT | Computed tomography |

dREV | Deterministic REV |

FIB | Focused ion beam |

LBM | Lattice Boltzmann method |

lREV | Local REV |

PN | Pore network |

PNM | Pore network modeling |

REA | Representative elementary area |

REV | Representative elementary volume |

SEM | Scanning electron microscope |

sREV | Statistical REV |

TPC | Two-point correlation |

## Appendix A. Penalty for the Pore Size Distribution

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**Figure 1.**A schematic representation of the striding-window selection in 2D for an image with a size of 2000 × 2000 voxels and a window size of 300 × 300 voxels. The step size is 100 voxels (grid lines) for illustration purposes. In practice, smaller step sizes are recommended. For illustration purposes, only some exemplary windows are shown. However, when applying the workflow, the full image is covered.

**Figure 2.**TPCs for all 130 image slices (solid opaque lines) and their means (dashed line). From top to bottom: pore space (phase zero, blue), AM (phase one, orange), and CBD (phase two, green). The convergence of all functions is clearly visible.

**Figure 3.**Deviation between the local and global results for the structural influencing factors. In addition, results for the penalty function are given. Values are shown as heatmaps in the range indicated by the legend and color map. Each pixel corresponds to one window (details in text).

**Figure 4.**LBM simulation results for ${D}_{\mathrm{eff}}/D$ for the four different cut sizes (symbols) compared to the value for the full image (dashed lines).

**Figure 5.**LBM simulation results for the permeability k for the four different cut sizes (symbols) compared to the value for the full image (dashed line).

**Figure 6.**Total transferred charge in charging simulations for the different cut sizes (symbols). For comparison, the median (dashed line), min–max range (lighter shaded area), and the standard deviation range around the mean (darker shaded area) for the nine largest simulations are shown (details in text).

Porosity ${\mathit{\varphi}}_{0}$ (%) | Binder Vol. Frac. ${\mathit{\varphi}}_{2}$ (%) | Tortuosity $\mathit{\tau}$ (-) | Spec. Surf. Area S (1/m) |
---|---|---|---|

21.09 | 13.02 | 1.2 | 188,209 |

${\mathit{D}}_{\mathbf{eff}}^{\mathbf{\left(}\mathbf{void}\mathbf{\right)}}/\mathit{D}$ (-) | ${\mathit{D}}_{\mathbf{eff}}^{\mathbf{\left(}\mathbf{solid}\mathbf{\right)}}/\mathit{D}$ (-) | k (1/m${}^{2}$) |
---|---|---|

0.085 | 0.690 | $9.89\times {10}^{-13}$ |

**Table 3.**Electrochemical simulation results for total transferred charge simulated with sub-images of shape $(666,\phantom{\rule{0.166667em}{0ex}}130,\phantom{\rule{0.166667em}{0ex}}666)$ to determine properties of the full image.

Median (mAh/cm${}^{2}$) | min|max (mAh/cm${}^{2}$) | Mean ± Std. Deviation (mAh/cm${}^{2}$) |
---|---|---|

1.840 | 1.813|1.864 | $1.836\pm 0.017$ |

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## Share and Cite

**MDPI and ACS Style**

Kellers, B.; Lautenschlaeger, M.P.; Rigos, N.; Weinmiller, J.; Danner, T.; Latz, A.
Systematic Workflow for Efficient Identification of Local Representative Elementary Volumes Demonstrated with Lithium-Ion Battery Cathode Microstructures. *Batteries* **2023**, *9*, 390.
https://doi.org/10.3390/batteries9070390

**AMA Style**

Kellers B, Lautenschlaeger MP, Rigos N, Weinmiller J, Danner T, Latz A.
Systematic Workflow for Efficient Identification of Local Representative Elementary Volumes Demonstrated with Lithium-Ion Battery Cathode Microstructures. *Batteries*. 2023; 9(7):390.
https://doi.org/10.3390/batteries9070390

**Chicago/Turabian Style**

Kellers, Benjamin, Martin P. Lautenschlaeger, Nireas Rigos, Julius Weinmiller, Timo Danner, and Arnulf Latz.
2023. "Systematic Workflow for Efficient Identification of Local Representative Elementary Volumes Demonstrated with Lithium-Ion Battery Cathode Microstructures" *Batteries* 9, no. 7: 390.
https://doi.org/10.3390/batteries9070390